Artin–Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin–Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin–Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin–Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.